Eisenstein's Criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers: if some prime divides every coefficient except the leading one, does not divide the leading coefficient, and its square does not divide the constant term, the polynomial cannot be factored into lower-degree polynomials with rational coefficients. Named for Ferdinand Eisenstein, it is a standard tool for proving irreducibility without directly searching for factorizations.
Facts
StatementIf there exists a prime number p such that p divides every coefficient of the polynomial except the leading one, p does not divide the leading coefficient, and p squared does not divide the constant term, then the polynomial is irreducible over the rational numbers. 1 Proof Method Classification
Statement Form Connections
Has Statement Form
In Branch
Source Eisenstein's criterion - Wikipedia
Principal Result Of
Source Eisenstein's criterion - Wikipedia
Proved By
Source Eisenstein's criterion - Wikipedia
Sources
1. Eisenstein's criterion - Wikipedia
History
Subsequently, Eisenstein published a somewhat different version in 1850, also in Crelle's Journal.
History section
Subsequently, Eisenstein published a somewhat different version in 1850, also in Crelle's Journal.
Principal Result Of: Gotthold Eisenstein, Lead sentence of the article on the result
This criterion is named after Gotthold Eisenstein.
In Group: Polynomials and Elementary Algebra, Lead section
Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over
In Branch: Algebra, Lead section
Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers -
Proved By: Gotthold Eisenstein, Lead
This criterion is named after Gotthold Eisenstein.
View the Source2. ProofWiki: Schönemann-Eisenstein Theorem
ProofWikiProof sectionQuote, Proof section
Aiming for a contradiction, suppose that $f = g h$ where $g, h \in \Z \sqbrk x$ are both non-constant.
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